social’and’technological’’ network’analysis’ lecture’7:’epidemics...
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Social and Technological Network Analysis
Lecture 7: Epidemics Spreading
Dr. Cecilia Mascolo
In This Lecture
• In this lecture we introduce the process of spreading epidemics in networks.
– This has been studied widely in biology. – But it also has important parallels in informaEon/idea diffusion in networks.
Epidemics vs Cascade Spreading
• In cascade spreading nodes make decisions based on pay-‐off benefits of adopEng one strategy or the other.
• In epidemic spreading – Lack of decision making. – Process of contagion is complex and unobservable • In some cases it involves (or can be modeled as randomness).
Branching Process
• Simple model. • First wave: A person carrying a disease enters the populaEon and transmit to all he meets with probability p. He meets k people: a porEon of which will be infected.
• Second wave: each of the k people goes and meet k different people. So we have a second wave of kxk=k2 people.
• Subsequent waves: same process.
Example with k=3
High contagion probability: The disease spreads
Low contagion probability: The disease dies out
Basic ReproducEve Number
• Basic ReproducEve Number R0=p*k – It determines it the disease will spread or die out.
• In the branching process model, if R0<1 the disease will die out aYer a finite number of waves. If R0>1, with probability >0, the disease will persist by infecEng at least one person in each wave.
Measures to limit the spreading
• When R0 is close 1, slightly changing p or k can result in epidemics dying out or happening. – QuaranEning people/nodes reduces k. – Encouraging be]er sanitary pracEces reduces germs spreading [reducing p].
• LimitaEons of this model: – No realisEc contact networks: no triangles! – Nodes can infect only once. – No nodes recover.
Formal Epidemics Models The SI Model
• S: suscepEble individuals. • X: infected individuals, when infected they can infect others conEnuously (different from before).
• n: total populaEon. • β (called k before) is the number of contacts per unit of Eme of an individual.
• SuscepEble contacts per unit of Eme βS/n. • Overall rate of infecEon XβS/n.
SuscepEble(S)
Infected (X)
SI Model
dXdt
= !SXn
dSdt
= !!SXn
s = Sn
!
x =Xn
!
s =1" xdxdt
= #(1" x)x
Logis6c Growth Equa6on
Eme
infected
x(t) = x0e!t
1! x0 + x0e!t
SIR Model
• Infected nodes recover at a rate γ. • A node stays infected for τ Eme. • Branching process is SIR with τ=1.
SuscepEble(S)
Infected (I)
Removed (R)
!
dsdt
= "#sx
dxdt
= #sx " $x
drdt
= $x
s+ x + r =1
Example
• The soluEon to the system is complex • Numerical examples of soluEon: • β=1, γ=0.4, s(at start)=0.99, x(at start)=0.01, r(at start)=0
FracEo
n of nod
es
Infected
recovered suscepEble
Eme
Epidemic Threshold
• When would the epidemic develop and when would it die out?
• It depends on the relaEonship of β and γ: – Basic ReproducEve Number R0=β/γ – If the infecEon rate [per unit of Eme] is higher than the removal rate the infecEon will survive otherwise it will die out.
– In SI, γ=0 so the epidemics always happen.
LimitaEons of SIR
• Contagion probability is uniform and “on-‐off”
• Extensions
– Probability q of recovering in each step. – Infected state divided into intermediate states (early, middle and final infecEon Emes) with varying probability during each.
– We have assumed homogenous mixing : assumes all nodes encounter each others with same probability: we could assume different probability per encounter.
SIS Model
SuscepEble(S)
Infected (X)
!
dsdt
= "x # $sx
dxdt
= $sx # "x
s+ x =1dxdt
= ($ # " # $x)x
• If β > γ growth curve like in SI but never reaching all populaEon infected. The fracEon of infected-‐>0 as β approaches γ. • If β< γ the infecEon will die out exponenEally. • SIS has the same R0 as SIR.
Relaxing AssumpEons
• Homogeneous Mixing: a node connects to the same average number of other nodes as any other.
• Most real networks are not random networks where the homogeneous mixing assumpEon holds.
• Most networks have different degree distribuEons. – Scale free networks!
Would the model apply to SF?
• Pastor-‐Satorras and Vespignani [2001] have considered the life of computer viruses over Eme on the Internet:
Surviving probability of virus Virus survived on average 6-‐9/14 months depending on type
How to jusEfy this survival Eme?
• The virus survival Eme is considerably high with respect to the results of epidemic models of spreading/recovering: – Something wrong with the epidemic threshold!
• Experiment: SIS over a generated Scale Free network (exponent -‐3).
No Epidemic Threshold for SF!
Random SF
threshold
Random Network
Scale Free Network InfecEons proliferate in SF networks independently of their spreading rates!
Following result on ImmunizaEon
• Random network can be immunized with some sort of uniform immunizaEon process [oblivious of the characterisEcs of nodes].
• This does not work in SF networks no ma]er how many nodes are immunized [unless it is all of them].
• Targeted immunizaEon needs to be applied – Keeping into account degree!
ImmunizaEon on SF Networks
• Red=SF • Black= Random
Uniform ImmunizaEon
Targeted ImmunizaEon Uniform and Targeted ImmunizaEon
SIRS Model
• SIR but aYer some Eme an R node can become suscepEble again.
• A number of epidemics spread in this manner (remaining latent for a while and having bursts).
SuscepEble(S)
Infected (I)
Refractory (R)
ApplicaEon of SIRS to Small World Models
Numerical Results
• c is the jumping probability
Summary
• Epidemics are very complex processes. • ExisEng models have been increasingly capable of capturing their essence.
• However there are sEll a number of open issues related to the modelling of real disease spreading or informaEon disseminaEon.
References • Chapter 21 • Pastor-‐Satorras, R. and Vespignani, A. Epidemic Spreading in Scale-‐Free Networks. Phys. Rev. Le].(86), n.14. Pages = 3200-‐-‐3203. 2001.
• Pastor-‐Satorras, R. and Vespignani, A. ImmunizaEon of Complex Networks. Physical Review E 65. 2002.
• Marcelo Kuperman and Guillermo Abramson. Small world effect in an epidemiological model. Physical Review Le]ers, 86(13):2909–2912, March 2001.